Well-posedness and Scattering for the Boltzmann Equations: Soft Potential with Cut-off
Well-posedness and Scattering for the Boltzmann Equations: Soft Potential with Cut-off
复制标题
玻尔兹曼方程的适定性和散射：带截止的软势
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Jin
中科院分区：
文献类型：
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作者：
Lingbing He;Jin
本文证明了软势模型γ = 2-N的截断Boltzmann方程Cauchy问题的唯一温和解的整体存在性，其中N= 2，3是维数.该证明依赖于Ovcharov（SIAM J Math Anal 43（3）：1282-1310，2011）对动力学方程的现有非齐次的Schahartz估计和Alonso等人（Commun Math Phys 298：293-322，2010）对玻尔兹曼碰撞算子的增益项的卷积类估计。整体动力学的解决方案的特点也表明，小的全球解决方案散射的动力学输运算子在$L^N_{x，v}$Lx，vN。对于具有大初值的Cauchy问题，给出了函数空间与截断软势模型$$-N<gamma <2-N$-N<γ<2-N之间的联系的局部适定性结果.
We prove the global existence of the unique mild solution for the Cauchy problem of the cut-off Boltzmann equation for soft potential model $$gamma =2-N$$γ=2-N with initial data small in $$L^N_{x,v}$$Lx,vN where $$N=2,3$$N=2,3 is the dimension. The proof relies on the existing inhomogeneous Strichartz estimates for the kinetic equation by Ovcharov (SIAM J Math Anal 43(3):1282–1310, 2011) and convolution-like estimates for the gain term of the Boltzmann collision operator by Alonso et al. (Commun Math Phys 298:293–322, 2010). The global dynamics of the solution is also characterized by showing that the small global solution scatters with respect to the kinetic transport operator in $$L^N_{x,v}$$Lx,vN. Also the connection between function spaces and cut-off soft potential model $$-N<gamma <2-N$$-N<γ<2-N is characterized in the local well-posedness result for the Cauchy problem with large initial data.